Maximizing Profits

[thunc14]

Problem: You set up a business making and selling books. The average hardcover book sells for $20 and the average paperback sells for $15. On average, a hardcover book costs $5 to make and a paperback costs $3 to make. If you can only spend $450 per day on books and you need to make at least 100 books per day, how many of each type should you make to maximize profit?
a) 150 paperback, 0 hardcover
b) 80 paperback, 60 hardcover
c) 25 paperback, 75 hardcover
d) 0 paperback, 100 hardcover
e) The answer is not listed

So B is out because the cost of that is $540, and D is out because that costs $500. A gives you a profit of $1800 and C gives you a profit of $1425. A seems like the answer, but it is possible the profit has not been maximized. My question is hot to prove that a maximum profit has been achieved?
My equations are:
H+P≥100
5H+3P≤450
and of course Profit = 15H+12P

When I graph the first 2 equations, the point of intersection is (75, 25), meaning 75 hard books and 25 paperback books. What am I doing wrong?

 

[MarkFL]

Consider the following graph:



Which of the 3 vertices maximizes the profit function?

 

[Steven G]

Problem: You set up a business making and selling books. The average hardcover book sells for $20 and the average paperback sells for $15. On average, a hardcover book costs $5 to make and a paperback costs $3 to make. If you can only spend $450 per day on books and you need to make at least 100 books per day, how many of each type should you make to maximize profit?
a) 150 paperback, 0 hardcover
b) 80 paperback, 60 hardcover
c) 25 paperback, 75 hardcover
d) 0 paperback, 100 hardcover
e) The answer is not listed

So B is out because the cost of that is $540, and D is out because that costs $500. A gives you a profit of $1800 and C gives you a profit of $1425. A seems like the answer, but it is possible the profit has not been maximized. My question is hot to prove that a maximum profit has been achieved?
My equations are:
H+P≥100
5H+3P≤450
and of course Profit = 15H+12P

When I graph the first 2 equations, the point of intersection is (75, 25), meaning 75 hard books and 25 paperback books. What am I doing wrong?

You did not list ALL the equations. Maybe if you did, you would have the same 3 points that MarkFL got. Do you know the two equations which you did not include?

 

[thunc14]

H≥0 and P≥0? Only thing I can think of unless I graphed the revenue from the books and the cost of making them separately, rather than together as profit.

 

[Steven G]

H≥0 and P≥0? Only thing I can think of unless I graphed the revenue from the books and the cost of making them separately, rather than together as profit.

Yes, now include those inequalities into your graph and see if you get additional corner points. Then reexamine the graph that MarkFL posted.